The Chain Rule

(This topic is also in Section 4.4 in Applied Calculus 5e and Section or Section 11.4 of Finite Mathematics and Applied Calculus 5e)

Warmup Let us start with a quick quiz on using the "Calculation Thought Experiment (CTE)" discussed in the preceding tutorial:

Calculation Thought Experiment (CTE)

The calculation thought experiment is a technique to determine whether to treat an algebraic expression as a product, quotient, sum, difference, power...:

    Given an expression, consider the steps you would use in computing its value. If the last operation is multiplication, treat the expression as a product; if the last operation is division, treat the expression as a quotient; if the last operation is raising to a power, treat the expression as a power, and so on.

Using the Calculation Thought Experiment (CTE) to Differentiate a Function

    If the CTE says, for instance, that the expression is a sum of two smaller expressions, then apply the rule for sums as a first step. This will leave you having to differentiate simpler expressions, and you can use the CTE on these, and so on...


Examples

1. (3x2-4)(2x+1) can be computed by first calculating the expressions in parentheses and then multiplying. Since the last step is multiplication, we can treat the expression as a product.

2. (2x-1)/x can be computed by first calculating the numerator and denominator, and then dividing one by the other. Since the last step is division, we can treat the expression as a quotient.

3. x2 + (4x-1)(x+2) can be computed by first calculating x2, then calculating the product (4x-1)(x+2), and finally adding the two answers. Thus, we can treat the expression as a sum.

4. (3x2-1)5 can be computed by first calculating the expression in parentheses, and then raising the answer to the fifth power. Thus, we can treat the expression as a power.


 
Q The expression  
3x-1

x2-ex
  -4 is written as

 
Q The expression (2x-5)-1(x2+3x-1)-1.5 is written as

Q Now that we are done with the preliminaries, what is the chain rule?
A Here is an example: We know that the derivative of x3 is 3x2. What, then, would you say is the deriviative of something more complicated raised top the third power, for instance (2x + x-1.4)3 ?

Q Is it not just 3(2 + 1.4x0.4)2 ?
A No. To find the correct answer, we use the chain rule.

The Chain Rule

If u is a differentiable function of x, and f is a differentiable function of u, then f is a differentiable function of x, and:

    d

    dx
    [f(u)] = f'(u)
    du

    dx

Example

Taking f(x) = x3, we get

    d

    dx
    u3 = 3u2
    du

    dx

In words:

    The derivative of a quantity cubed is 3 times that (original) quantity squared, times the derivative of the quantity.

This is sometimes referred to as an example of the generalied power rule.

More Examples
1.
d

dx
(1+x2)3 =
3(1+x2)2
d

dx
(1+x2)The derivative of a quantity cubed = 3 times that (original) quantity squared, times the derivative of the quantity.
=
3(1+x2)2,2x
=
6x(1+x2)2
2.
d

dx
2

(x+x2)3
=
d

dx
2(x+x2)-3
=
2(-3)(x+x2)-4
d

dx
(x+x2)The derivative of a quantity to the -3 = -3 times that (original) quantity to the -4, times the derivative of the quantity.
=
-6(x+x2)-4 (1+2x)
=
-6(1+2x)

(x+x2)4

 
Q Why is the chain rule true?
A Press here for a proof.

Here are some for you to try. Note Use proper graphing calculator format to input your answers (spaces are ignored).

d

dx
(3x2-4)3
=      
d

dx
4

(x2+x)2
=      
d

dx
2x3 - x
=      

Q What about things other than powers -- things such as ln(x2 + 4x), for instance?
A The following table shows how we apply the chain rules to a whole variety of functions, but the derivatives of these functions appear later, so have patience, or follow the links below.

Original Rule
Generalized Rule
(Chain Rule)
Example
Comments
d

dx
f(x) = g(x)
d

dx
f(u) = g(u)
du

dx
General form of Chain Rule
d

dx
xn = nxn-1
d

dx
un = nun-1
du

dx
d

dx
5(4x + 3)0.5=2.5(4x + 3)-0.5(4)
=10(4x + 3)-0.5
Generalized Power Rule
d

dx
x=
1

2x
d

dx
u=
1

2u
du

dx
d

dx
1 + x2=
1

21 + x2
2x
=
x

1 + x2
Power rule with n = 1/2
d

dx
ex = ex
d

dx
eu = eu
du

dx
d

dx
e(3x-1)=e(3x-1)(3)
=3e(3x-1)
See next tutorial.
d

dx
sin x = cos x
d

dx
sin u = cos u
du

dx
d

dx
sin(3x-1)=cos(3x-1) (3)
=3cos(3x-1)
Take me to text on trig functions!

d

dx
 
3x - 1

x2 - x-1
  5 = ?

d

dx
[(x2 - 1)3(3x + 4)-1] = ?

Try some exercises for this topic here. Alternatively, try some of the exercises in Section 4.4 in Applied Calculus or Section 11.4 in Finite Mathematics and Applied Calculus.

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Last Updated: October 2009
Copyright © 2007 Stefan Waner